Energy forms on conformal <i>C</i><sup>1</sup>‐diffeomorphic images of the Sierpinski gasket

Freiberg, Uta; Lancia, Maria Rosaria

doi:10.1002/mana.200510606

Cited by 5 publications

(2 citation statements)

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“…Convergence schemes for Dirichlet forms on varying spaces of finite dimension have been considered by Mosco and others, particularly in the context of stochastic diffusion equations and diffusion on fractals, see e.g. [19,11,14,5,28]. There are similar convergence results for manifolds, metric measure spaces, Hilbert spaces, quadratic forms on different Hilbert spaces in [21,16,17].…”

Section: Introductionmentioning

confidence: 85%

Norm convergence of sectorial operators on varying Hilbert spaces

Mugnolo¹,

Nittka²,

Post³

2013

Operators and Matrices

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Abstract. Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on varying spaces is natural. Many previous contributions to this subject consider either concrete examples of perturbations, or an abstract setting where weak or strong convergence of the resolvents is used. However, it seems that the first results on norm resolvent convergence in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on Hilbert spaces that depend on a parameter. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces. In addition, we show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold's skeleton graph plays a prominent role, being our main application.Mathematics subject classification (2010): 34B45, 35P05, 47D06.

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Section: Introductionmentioning

confidence: 85%

Norm convergence of sectorial operators on varying Hilbert spaces

Mugnolo¹,

Nittka²,

Post³

2013

Operators and Matrices

View full text Add to dashboard Cite

show abstract

“…Since Goldstein and Kusuoka constructed Brownian motion on the Sierpinski gasket in Goldstein (1987) and Kusuoka (1989), diffusion processes on fractals and their associated Dirichlet form have been mostly studied for the case of self-similar sets (see e.g., Barlow and Perkins, 1988;Kigami, 1989Kigami, , 2001 while only certain deterministic non-self-similar cases have been considered yet (see e.g., Freiberg and Lancia, 2004, 2005, 2008.…”

Section: Introductionmentioning

confidence: 99%